Superposition principle

Coulomb’s law, as it stands, describes the force between only two charges. But what happens when many other charges act on a charge at the same time? The answer is as simple as it is powerful: the forces add up, and they add up vectorially.

Superposition principle

The resultant force on a charge $q$ is the vector sum of the forces that each of the other charges would exert on it individually: $\ev{\vv{F}_{\text{tot}} = \vv{F}_1 + \vv{F}_2 + \vv{F}_3 + \cdots}$

The crucial word is vectorially: each contribution $\vv{F}_i$ has its own magnitude (calculated with Coulomb’s law) and its own direction (along the line joining it to the $i$-th charge, attractive or repulsive according to the signs). The magnitudes are not summed; the components are: you must choose axes, project each force, sum the components along $x$ and along $y$ separately, and finally recompose the resultant vector.

This independence of the contributions — each pair of charges “unaware” of the others — is by no means obvious, but it is a property verified experimentally with very high precision. It is what makes systems with many charges tractable: however complicated the configuration, the force on each charge is always obtained as the sum of many simple Coulomb terms.

The typical application is calculating the force on a charge placed at the vertices of a geometric figure (see the worked exercise on three charges at the vertices of an equilateral triangle): the individual magnitudes are calculated, resolved along the axes exploiting symmetry, and the resultant is recomposed.

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Topics: Electrostatics Concepts: Coulomb’s law Skills: Superposition principle

Related exercises: Worked exercise — Three charges at the vertices of a triangle · Problem — Force between two 1-microcoulomb charges · Problem — Force between two spheres with the same charge